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Section 10-4
Pages 578-586
Inscribed Angles
After today’s lesson you should be able to:
- Find measures of inscribed angles and measures of angles
of inscribed polygons.
Inscribed Angles- an angle that has its vertex on the circle and its
sides contained in chords of the circle.
B
Vertex B is on the circle.
AB and BC are chords on
the circle.
A
D
C
Arc ADC is the arc intercepted by LABC.
Theorem 10.5: Inscribed Angle Th.- if an angle is inscribed in a
circle, then the measure of the angle equals one-half the measure of
its intercepted arc (or the measure of the intercepted arc is twice
the measure of the inscribed angle).
B
A
mLABC = ½(mLADC)
2(mLABC) = mLADC
D
C
In Circle F, mUW = 20, mXY = 40, mUZ = 108, and
mUW = mYZ. Find the measures of the numbered angles.
W
X
3
Y
4
5
2
U
1
Z
T
Theorem 10.6: If two inscribed angles of a circle (or congruent
circles) intercept congruent arcs or the same arc, then the angles
are congruent.
C
Inscribed L’s  if arcs are .
Inscribed L’s  if same arc intercepted.
Proof: Given- Circle X with CD  AB.
Prove- AXB  CXD
Statements
1. LDAB intercepts DB.
LDCB intercepts DB.
2. LDAB  LDCB.
3. L1  L2
4. CD  AB
5. AXB  CXD
A
X
D
B
Reasons
1. Def. of intercepted arc.
2. Inscribed L’s same arc 
3. Vert. L’s .
4. Given
5. AAS
Probability: Points M and N are on a circle so that mMN = 72.
Suppose point L is randomly located on the same circle so that it
does not coincide with M or N. What is the probability that
mLMLN = 144?
Theorem 10.7: if the inscribed angle of a triangle intercepts a
semicircle, the angle is a right angle.
A
ADC is a semicircle, so mLABC = 90.
B
D
C
Ex. 4: Triangles TVU and TSU are inscribed in Circle P with
VU  SU. Find the measure of each numbered angle if
mL2 = x + 9 and mL4 = 2x + 6.
U
V
S
T
Theorem 10.8: If a quadrilateral is inscribed in a circle, then its
B
opposite angles are supplementary.
ABCD is inscribed in circle P.
LA and LC are supplementary.
LB and LD are supplementary.
P
C
A
D
Ex. 5: Quadrilateral QRST is inscribed in circle M. If mLQ = 87
and mLR = 102, find mLS and mLT.
R
S
Q
T